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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.33
Chapter 4, Problem 4.7.33

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(t√t + √t) / t² dt

Verified step by step guidance
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First, simplify the integrand by expressing all terms with exponents. Recall that \( \sqrt{t} = t^{1/2} \) and rewrite the expression inside the integral accordingly.
Rewrite the integrand \( \frac{t\sqrt{t} + \sqrt{t}}{t^2} \) as \( \frac{t \cdot t^{1/2} + t^{1/2}}{t^2} = \frac{t^{3/2} + t^{1/2}}{t^2} \).
Split the fraction into two separate terms: \( \frac{t^{3/2}}{t^2} + \frac{t^{1/2}}{t^2} \). Then simplify each term by subtracting exponents in the denominator from those in the numerator.
After simplification, express the integrand as a sum of powers of \( t \): \( t^{3/2 - 2} + t^{1/2 - 2} = t^{-1/2} + t^{-3/2} \).
Now, integrate each term separately using the power rule for integration: \( \int t^n dt = \frac{t^{n+1}}{n+1} + C \), making sure to add the constant of integration at the end.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
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Introduction to Indefinite Integrals

Algebraic Simplification of the Integrand

Before integrating, simplify the integrand by combining like terms and rewriting expressions using exponent rules. For example, rewrite roots as fractional exponents and simplify fractions to make integration straightforward and reduce errors.
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Completing the Square to Rewrite the Integrand

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is fundamental for integrating polynomial and power functions, allowing direct computation once the integrand is expressed in terms of powers.
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Power Rule for Indefinite Integrals
Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = (x² − 3) / (x − 2), x ≠ 2

Textbook Question

Applications


Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?

Textbook Question

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


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Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ cos³ 𝓍/2 d𝓍

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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∫ ( 3√ t + 4/t² ) dt